Fundamentals研发培训,设计培训

ReviewofFundamentals

HaranKarmakerAugust4,2005ReviewofFundamentals1Topics

VectorsandScalarsPartialDifferentialEquationsElectromagneticConceptsMaxwell’sEquationsElectricandMagneticCalculationsPrinciplesofMachineOperationMathematicalModelingMagneticFieldsTwo-ReactionTheoryReviewofFundamentals2VectorsandScalars

Vectorsandscalarsaremathematicalrepresentationsofphysicalquantities.Vectorshavebothmagnitudesanddirections.Scalarshaveonlymagnitudes,nodirections.Forexample,locationofapointinspacefromtheoriginisdefinedbyapositionvectorwiththreecomponentsalongthex,yandzaxes.Thedistanceofthepointfromtheoriginisascalarquantitywithoutanydirection.ReviewofFundamentals3VectorRepresentation

Intheexample,thefluxdensityintheairgapwillhavecomponentsintheradial,tangentialandaxialdirectionsandisavector.Theairgapfluxoveranareaisascalar.Anotherexampleofascalaristheroomtemperaturewhichhasnodirection.Thegradientofascalaristherateofchangeofthescalaralongadirection.Thegradientisavectorquantity.ReviewofFundamentals4VectorRepresentation

Thedivergenceofavectorquantityrepresentsthenetflowofthequantitythroughavolume.Thedivergenceisascalarfunctionofavectorfield.Thecurlofavectorisameasureofcirculationofthevectorfield.Stoke’stheoremrelatestheintegralofavectorfieldalongaclosedpathtotheintegralofthecurlofthefieldonthesurfacedefinedbytheclosedpath.Thethreevectorfunctions,gradient,divergenceandcurldescribethenatureofvariationofallphysicalquantitiesintheuniverse.ReviewofFundamentals5PartialDifferentialEquations

Thenatureofdistributionofaphysicalphenomenonisgovernedbyequationsincludingthechangeinmultiplevariablescalledpartialdifferentialequations.ThemostcommongoverningequationsareLaplace,Poisson,diffusionandwaveequations.Solutionofmostengineeringproblemscanbeformulatedasboundaryvalueproblems,whichrequiregoverningequationsandboundaryconditionsfortheirsolutions.Oncetheboundaryvalueproblemhasbeenformulated,itcanbesolvedbyanalyticalornumericalmethods.ReviewofFundamentals6ElectromagneticConcepts

Electrostaticfieldsarecausedbystationaryelectriccharges.In1785,Coulombinvestigatedthenatureofforcebetweentwochargedbodiesandformulatedthefollowingequationfromexperiments. F=Q1*Q2/(4*π*ε*r2) where F=ForceinNewtons Q1,Q2=ChargesinCoulomb ε=Permittivityofthemediuminfarads/m r=DistancebetweenchargesinmetersReviewofFundamentals7ElectromagneticConcepts

ElectricfieldintensityduetoachargeQisdefinedas E=Q/(4*π*ε*r2)ElectricfieldintensityisavectorwhosemagnitudeisinunitsofNewtonsperCoulomb,whichcanbeconvertedtotheunitsofvolts/meter.Therefore,thevectorEcanbetreatedasaforcefieldthatactsonacharge.Itcanalsobetreatedasagradientofavoltage.

ReviewofFundamentals8ElectromagneticConcepts

Electricfluxdensityisdefinedas D=εE IthasunitsofchargeperareaorCoulombspersquaremeter.AccordingtoGauss’slaw,theintegralofelectricfluxdensityoveraclosedsurfaceisequaltothefreechargeenclosedbythesurface.

ReviewofFundamentals9ElectromagneticConcepts

Thegoverningequationsforelectricfieldproblemsareoftendescribedbyapotentialfunctiondefinedas E=-VMathematicalrepresentationofGauss’slawgives .D=ρ whereρischargedensity.Therefore,- .(εV)=ρ V=-ρ/εPoisson’sequation V=0Laplace’sequation22ReviewofFundamentals10

Faraday’sLaw

Faraday’sLawstatesthatachangingmagneticfieldwillinduceanelectricfield.Theelectricfieldexistsinspaceregardlessofwhetheraconductorispresentornot.Whenaconductorispresent,acurrentwillflow.ThedifferentialformofFaraday’sLawisReviewofFundamentals11

Faraday’sLaw

InducedvoltagearoundastationaryclosedcontourClinkedbyachangingmagneticfieldisgivenbythelineintegralofelectricfield

ThemagneticfluxisTheintegralformofFaraday’sLawisReviewofFundamentals12

Ampere’sLaw

AvectorcalledmagneticfieldintensityisdefinedasThedifferentialformofAmpere’sLawisReviewofFundamentals13

Ampere’sLaw

IntegralformofAmpere’sLawisobtainedbyapplyingStoke’stheoremAmpere’sLawinintegralformstatesthatthelineintegraloffieldintensityaroundacontourisequaltothenetcurrentenclosedbythecontour.ReviewofFundamentals14

Maxwell’sEquations

Maxwell’sequationsdescribethetheoryofelectromagnetismbyunifyingalllaws.

divD=ρ

divB=0D=εEReviewofFundamentals15SymbolsinMaxwell’sEquations

PermeabilityisaphysicalpropertyofamaterialrelatingfluxdensityBtofieldintensityH.Permeabilityoffreespaceis4Пx1E-7Henry/m.Formagneticsteel,permeabilityvarieswithfluxdensityorfieldintensity.Electricconductivityisreciprocalofresistivityandvarieswithtemperature.ReviewofFundamentals16ElectricandMagneticCalculations

Formanymagneticapplications,goodapproximatesolutionscanbeobtainedbyacircuitanalysissimilartothatofad.c.circuitcomposedofseriesandparallelcombinationsofresistors.Forexample,consideratoroidwithNturnscarryingcurrentI.ReviewofFundamentals17ElectricandMagneticCalculations

Themagneticfieldintensityinthetoroidiscontinuous.Thefluxdensityismuchgreaterinsidethanoutsidebecauseofthepermeabilityofthemagneticmaterial.ReviewofFundamentals18ElectricandMagneticCalculations

ApplyingAmpere’sLawaroundthecircularpathCintheinterioroftoroid, H=N*I/(2*π*d) Amps/m whereN*IiscalledtheMMF(MagnetoMotiveForce)analogoustoEMF(ElectroMotiveForceorVolts)inelectriccircuit.Similarly,magneticfluxisanalogoustoelectriccurrent.ReviewofFundamentals19ElectricandMagneticCalculations

Ohm’sLawforelectriccircuit, E=I*RwhereR=electricresistance=l/(ρ*A)Similarly,formagneticcircuit, MMF=Φ*RwhereR=magneticreluctance=l/(μ*A)ReviewofFundamentals20ElectricandMagneticCalculations

Tounderstandtheconceptofinductance,considertwocircuitsmagneticallycoupled.Anychangeincurrentresultsinachangeinmagneticfield.Whencurrentchanges,thefluxlinkingthecircuitschangeandvoltagesareinducedinthecircuits.

ReviewofFundamentals21ElectricandMagneticCalculations

Theselfinductanceisdefinedastheratiooffluxlinkingthecircuittothecurrentinthesamecircuit. L11=Φ11/I1Themutualinductanceisdefinedasthefluxlinkingthecircuitbyasecondcircuittothecurrentinthesecondcircuit. L12=Φ12/I2

ReviewofFundamentals22MagneticForces

ThemagneticforceonaconductoroflengthLinmagneticfieldBis F=I*LxB NewtonsTheforcedensityonaconductoris

F=JxB Newtons/m^3 whereJisthecurrentdensity,Amps/m^2

ReviewofFundamentals23PrinciplesofMachineOperation

IfanNturncoilislinkedbyafluxwhosetimerateofchangeisdF/dT,theterminalsofthecoilwillhaveavoltageinducedaccordingtoFaraday’sLawReviewofFundamentals24PrinciplesofMachineOperation

Consideraseriesofmagnetsmovingpastacoil.Aseachmagnetmovespastthecoil,thecoilseesanincreaseoffluxlinkage,henceanincreaseofvoltagefollowedbyadecreaseoffluxlinkagehenceadecreaseofvoltage.Ifwefurtherarrangeitsuchthatthemagnetsalternatepolarity,theresultingvoltagewillriseabove“zero”voltstoamaximumanddecreasebelow“zero”voltstoaminimum.ReviewofFundamentals25PrinciplesofMachineOperation

Considernowanotherstationarycoilfixedadjacenttothefirstone.Anothersimilaralternatingvoltagewillappearonitsterminals.

ReviewofFundamentals26PrinciplesofMachineOperation

Thewaveshapeofthesecondvoltagewillbeidenticaltothefirstcoil,howeverthetimingwillbedifferent.Theamountofthis“phasedifference”willdependonthespeedofthemagneticpolesandonthedistancebetweenthetwostationarycoils.ReviewofFundamentals27PrinciplesofMachineOperation

FundamentalFluxandVoltageinamachineisgivenbyReviewofFundamentals28SymbolsinVoltageEquation

Φf=fundamentalfluxperpole(Wb)kVll=line-linekVN=numberofturnsinseriesperphasef=frequency,HzKp=pitchfactorKd=distributionfactorKs=skewfactorReviewofFundamentals29SymbolsinVoltageEquation

PitchfactorinvoltageequationisdefinedasnisharmonicorderandPUPisperunitpitch(ratioofspantopolepitch)ReviewofFundamentals30SymbolsinVoltageEquation

Distributionfactorinvoltageequationis

numberofcoils(slots)perpoleperphasereducedtolowestterm

A+B/C

ReviewofFundamentals31SymbolsinVoltageEquation

Skewfactorinvoltageequationis

skewangleλisshownbelow

ReviewofFundamentals32SlotCombinations

Theterm‘slots’whenusedinelectricaldesignisgenerallyinterchangeablewithcoils.Sinceeachslothastwocoillegs,thenumberofcoilsandthenumberofslotsareidentical.Thebasicartofthedesignofmachinesisthatofselectingthenumberofcoils,turnsandcircuitstoprovidetheoptimumdesign.Foreconomicreasons,allcoilsareusuallymadeidentical,thatisthesamenumberofturnspercoil,thesamespan,andthesameskew.ReviewofFundamentals33SlotCombinations

ConsiderawindingofNppoles,andNscoils.Thenumberofcoilsperpoleperphase

Forexample,324slots,24polesand3

phasesgivesReviewofFundamentals34SlotCombinations

Thisistheaveragenumberofcoilsofonephaseundereachpole.Tooptimizethedistribution,theactualnumberundereachpolewillvarywiththepole,butwillbeascloseaspossibletothisnumber.Henceinthiscase,pole#1mayhave4phase‘a’coilswhilepole#2mayhave5phase‘a’coilsandsoonaroundtheunit.ReviewofFundamentals35SlotCombinations

Thepatternofphase‘a’coilswithrespecttopolenumberwouldbeReviewofFundamentals36SlotCombinations

Thenumberofcoilsperpole

Inthisexample,D+E/Fis13+1/2.Weput13coilsunderonepoleand14underthenextpole.ReviewofFundamentals37SlotCombinations

Thewindingpatternisthesequenceofnumbersthatrepresentsthephase‘a’,‘b’and‘c’coilsastheyarelaidout.Inthisexample

ReviewofFundamentals38CoilsandPolesPerCircuit

Eachcircuitshouldhavethesamenumberofcoilsandshouldcoveranintegralnumberofpoles.Theseassumptionsmaybeviolatedresultinginanunbalancedwinding.Circuitsarelaidontothewindingpatternafterthepatternhasbeensketchedaroundthemachine.

ReviewofFundamentals39SingleandDoubleLayerWindings

Singlelayerwindingshaveeacharmatureslotcontainingonlyonecoilleg.Doublelayerwindingshavetwolegsperslotandaremostcommonforlargemachines.

ReviewofFundamentals40ArmatureReaction

Besidesthemagneticfieldsgeneratedbytherotorpoles,additionalfieldsarecausedbythecurrentsinthestatorcoils.Thestatorcurrentssetupadistributionofmmf(withharmonics),thefundamentalofwhichrotatesatsynchronousspeed.Armaturereactionaffectstheairgapfluxinvariouswaysdependingonthewindingarrangement,thephasecurrents,thereluctanceofthemainfluxpathandthepowerfactor.

ReviewofFundamentals41Two-ReactionTheory

Theairgapfluxinasalientpolesynchronousmachineisdistortedduetothenon-uniformairgapsoverpolesurfaceandbetweenthepoles.Thephasordiagramsofsalientpolemachinesarerepresentedbysepara